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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40051, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([9,307]))
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gp:[g,chi] = znchar(Mod(10898, 40051))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40051.10898");
| Modulus: | \(40051\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(40051\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | yes |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{40051}(260,\cdot)\)
\(\chi_{40051}(501,\cdot)\)
\(\chi_{40051}(866,\cdot)\)
\(\chi_{40051}(1052,\cdot)\)
\(\chi_{40051}(1619,\cdot)\)
\(\chi_{40051}(1856,\cdot)\)
\(\chi_{40051}(2274,\cdot)\)
\(\chi_{40051}(2631,\cdot)\)
\(\chi_{40051}(2746,\cdot)\)
\(\chi_{40051}(2888,\cdot)\)
\(\chi_{40051}(3086,\cdot)\)
\(\chi_{40051}(3284,\cdot)\)
\(\chi_{40051}(3878,\cdot)\)
\(\chi_{40051}(4461,\cdot)\)
\(\chi_{40051}(4826,\cdot)\)
\(\chi_{40051}(5546,\cdot)\)
\(\chi_{40051}(5836,\cdot)\)
\(\chi_{40051}(6662,\cdot)\)
\(\chi_{40051}(6706,\cdot)\)
\(\chi_{40051}(6717,\cdot)\)
\(\chi_{40051}(6837,\cdot)\)
\(\chi_{40051}(7103,\cdot)\)
\(\chi_{40051}(8742,\cdot)\)
\(\chi_{40051}(9517,\cdot)\)
\(\chi_{40051}(9654,\cdot)\)
\(\chi_{40051}(10633,\cdot)\)
\(\chi_{40051}(10898,\cdot)\)
\(\chi_{40051}(11013,\cdot)\)
\(\chi_{40051}(11651,\cdot)\)
\(\chi_{40051}(12502,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\((11255,17546)\) → \((e\left(\frac{3}{110}\right),e\left(\frac{307}{330}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 40051 }(10898, a) \) |
\(1\) | \(1\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{109}{330}\right)\) | \(e\left(\frac{31}{165}\right)\) | \(e\left(\frac{94}{165}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{57}{110}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
